Lesson 0: The Big Picture — What Is Complex Analysis About?

Part 1: Domain coloring

The core problem

A real function $f:\mathbb{R}\to\mathbb{R}$ is a curve, graphable in 2D. A complex function $f:\mathbb{C}\to\mathbb{C}$ maps a plane to a plane: the input $z$ has 2 real coordinates, and so does the output $f(z)$. A literal graph of $(z, f(z))$ would need 4 real dimensions, not drawable. Complex analysis needs substitute visualization tools, and this lesson introduces two of them.

The bridge to vector calculus

Every complex function can be written $f(x+iy) = u(x,y) + iv(x,y)$, splitting it into two real-valued functions of two real variables. This is exactly the kind of $\mathbb{R}^2 \to \mathbb{R}^2$ map that shows up in vector calculus: vector fields, parametrized maps. This connection keeps resurfacing throughout the subject, especially once the Cauchy-Riemann equations enter the picture.

Tool 1: Domain coloring

Plot the input plane in 2D, but color each point $z$ by $\arg(f(z))$, the angle (argument) of the output value $f(z)$.

For nonzero $z = x+iy$, write $z = re^{i\theta}$ in polar form, where $r = |z|$ is the distance from the origin and $\theta = \arg(z)$ is the angle from the positive real axis, measured counterclockwise.

Worked example: $f(z) = z^2$

• In polar form, $z = re^{i\theta} \implies z^2 = r^2 e^{i2\theta}$, so $\arg(z^2) = 2\arg(z)$.
• Walking once around the origin ($\theta: 0 \to 2\pi$) makes $\arg(z^2)$ sweep $0 \to 4\pi$, so the color cycle repeats twice.
• Compare to $f(z) = z$ (the identity map): $\arg(f(z)) = \arg(z)$, a 1-to-1 correspondence, colors cycle exactly once.
• $z^2$ is therefore a 2-to-1 map in general: $z$ and $-z$ always satisfy $z^2 = (-z)^2$ (e.g. $1^2 = (-1)^2 = 1$).
• The only point where this 2-to-1 behavior collapses to 1-to-1 is where $z = -z$, i.e. $z = 0$, the unique fixed point of $z \mapsto -z$.

Key takeaways: $f(x+iy) = u(x,y) + iv(x,y)$ is the universal decomposition of a complex function into two real functions. $f(z)=z^2$ is 2-to-1 everywhere except $z=0$, where the two preimages $z, -z$ collide. That's the algebraic root cause of the "pinching" visible in the domain-coloring picture at the origin, and (as later lessons show) a sign that $f'(0) = 0$.

Part 2: Grid deformation and conformal maps

Worked example: $u(x,y), v(x,y)$ for $z^2$

Expanding $(x+iy)^2 = x^2-y^2+2ixy$ gives $u(x,y)=x^2-y^2$ and $v(x,y)=2xy$. This holds at every point — spot-checking $z=1,\ i,\ 1+i,\ 2+3i,\ 3-2i,\ -1+i$ all confirm $u+iv = z^2$.

The correspondence $(x,y)\leftrightarrow z=x+iy$ identifies $x$ with the real part and $y$ with the coefficient of $i$, the same identification of $\mathbb{C}$ with $\mathbb{R}^2$ used throughout complex analysis.

Grid deformation

A vertical line $x=c$ maps under $z^2$ to the curve

$$u = c^2 - \frac{v^2}{4c^2}$$

a parabola opening in the $-u$ direction, with vertex at $(c^2,0)$.

A horizontal line $y=c$ maps to

$$u = \frac{v^2}{4c^2} - c^2$$

a parabola opening in the $+u$ direction, with vertex at $(-c^2,0)$.

Conformality

A map $f$ is conformal at $z_0$ if it preserves the angle (magnitude and orientation) between any two curves crossing at $z_0$, once they're mapped to their images crossing at $f(z_0)$.

A worked check: at $z=1+i \to w=2i$, the images of the lines $x=1$ and $y=1$ are $u=1-\frac{v^2}{4}$ and $u=\frac{v^2}{4}-1$. At $v=2$, their slopes are $\frac{du}{dv}=-1$ and $+1$. The product of the slopes is $-1$, which means the curves are perpendicular, confirming conformality at this point.

Connection back to the pinch at $z=0$: at $c=0$, both parabola formulas above involve division by zero: conformality breaks down exactly where $f'(0)=2(0)=0$. This is a preview of a theorem proved rigorously in Lesson 2: a holomorphic function $f$ is conformal at $z_0$ if and only if $f'(z_0)\neq 0$.

Roadmap

Domain coloring and grid deformation are "what you see," the geometric picture. Lesson 1 (complex arithmetic, polar form, topology) and Lesson 2 (Cauchy-Riemann equations, the conformality proof) supply the "why." From there, the syllabus builds toward Cauchy's theorem and the Cauchy integral formula, the centerpiece of the subject, followed by residues, conformal mapping, and the Riemann mapping theorem.

Key takeaways: $u(x,y)=x^2-y^2,\ v(x,y)=2xy$ for $z^2$. Vertical lines map to left-opening parabolas; horizontal lines map to right-opening parabolas. A holomorphic $f$ is conformal at $z_0$ if and only if $f'(z_0)\neq 0$, exactly why $z^2$'s conformality fails at the origin.

Two pictures of a complex function

A complex function $f$ takes a complex number $z$ as input and produces a complex number $w=f(z)$ as output. Writing $z=x+iy$, the input is identified with the point $(x,y)$ in the input plane (the $z$-plane).

The output $w=f(z)$ can always be written as $w=u(x,y)+iv(x,y)$, where $u$ and $v$ are real-valued functions of the two real input variables $x,y$. This identifies the output with the point $(u,v)$ in the output plane (the $w$-plane).

So $f$ has two equivalent descriptions: as a single complex-valued map $z\mapsto f(z)$, or as a pair of real-valued maps $(x,y)\mapsto u(x,y)$ and $(x,y)\mapsto v(x,y)$, an $\mathbb{R}^2\to\mathbb{R}^2$ map $(x,y)\mapsto(u,v)$, the same kind of object as the vector-valued maps from vector calculus.